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## G = C2×C6×C12order 144 = 24·32

### Abelian group of type [2,6,12]

Aliases: C2×C6×C12, SmallGroup(144,178)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6×C12
 Chief series C1 — C2 — C6 — C3×C6 — C3×C12 — C6×C12 — C2×C6×C12
 Lower central C1 — C2×C6×C12
 Upper central C1 — C2×C6×C12

Generators and relations for C2×C6×C12
G = < a,b,c | a2=b6=c12=1, ab=ba, ac=ca, bc=cb >

Subgroups: 162, all normal (8 characteristic)
C1, C2, C2, C3, C4, C22, C6, C2×C4, C23, C32, C12, C2×C6, C22×C4, C3×C6, C3×C6, C2×C12, C22×C6, C3×C12, C62, C22×C12, C6×C12, C2×C62, C2×C6×C12
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C23, C32, C12, C2×C6, C22×C4, C3×C6, C2×C12, C22×C6, C3×C12, C62, C22×C12, C6×C12, C2×C62, C2×C6×C12

Smallest permutation representation of C2×C6×C12
Regular action on 144 points
Generators in S144
(1 128)(2 129)(3 130)(4 131)(5 132)(6 121)(7 122)(8 123)(9 124)(10 125)(11 126)(12 127)(13 65)(14 66)(15 67)(16 68)(17 69)(18 70)(19 71)(20 72)(21 61)(22 62)(23 63)(24 64)(25 87)(26 88)(27 89)(28 90)(29 91)(30 92)(31 93)(32 94)(33 95)(34 96)(35 85)(36 86)(37 78)(38 79)(39 80)(40 81)(41 82)(42 83)(43 84)(44 73)(45 74)(46 75)(47 76)(48 77)(49 139)(50 140)(51 141)(52 142)(53 143)(54 144)(55 133)(56 134)(57 135)(58 136)(59 137)(60 138)(97 111)(98 112)(99 113)(100 114)(101 115)(102 116)(103 117)(104 118)(105 119)(106 120)(107 109)(108 110)
(1 39 109 50 28 18)(2 40 110 51 29 19)(3 41 111 52 30 20)(4 42 112 53 31 21)(5 43 113 54 32 22)(6 44 114 55 33 23)(7 45 115 56 34 24)(8 46 116 57 35 13)(9 47 117 58 36 14)(10 48 118 59 25 15)(11 37 119 60 26 16)(12 38 120 49 27 17)(61 131 83 98 143 93)(62 132 84 99 144 94)(63 121 73 100 133 95)(64 122 74 101 134 96)(65 123 75 102 135 85)(66 124 76 103 136 86)(67 125 77 104 137 87)(68 126 78 105 138 88)(69 127 79 106 139 89)(70 128 80 107 140 90)(71 129 81 108 141 91)(72 130 82 97 142 92)
(1 2 3 4 5 6 7 8 9 10 11 12)(13 14 15 16 17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32 33 34 35 36)(37 38 39 40 41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80 81 82 83 84)(85 86 87 88 89 90 91 92 93 94 95 96)(97 98 99 100 101 102 103 104 105 106 107 108)(109 110 111 112 113 114 115 116 117 118 119 120)(121 122 123 124 125 126 127 128 129 130 131 132)(133 134 135 136 137 138 139 140 141 142 143 144)

G:=sub<Sym(144)| (1,128)(2,129)(3,130)(4,131)(5,132)(6,121)(7,122)(8,123)(9,124)(10,125)(11,126)(12,127)(13,65)(14,66)(15,67)(16,68)(17,69)(18,70)(19,71)(20,72)(21,61)(22,62)(23,63)(24,64)(25,87)(26,88)(27,89)(28,90)(29,91)(30,92)(31,93)(32,94)(33,95)(34,96)(35,85)(36,86)(37,78)(38,79)(39,80)(40,81)(41,82)(42,83)(43,84)(44,73)(45,74)(46,75)(47,76)(48,77)(49,139)(50,140)(51,141)(52,142)(53,143)(54,144)(55,133)(56,134)(57,135)(58,136)(59,137)(60,138)(97,111)(98,112)(99,113)(100,114)(101,115)(102,116)(103,117)(104,118)(105,119)(106,120)(107,109)(108,110), (1,39,109,50,28,18)(2,40,110,51,29,19)(3,41,111,52,30,20)(4,42,112,53,31,21)(5,43,113,54,32,22)(6,44,114,55,33,23)(7,45,115,56,34,24)(8,46,116,57,35,13)(9,47,117,58,36,14)(10,48,118,59,25,15)(11,37,119,60,26,16)(12,38,120,49,27,17)(61,131,83,98,143,93)(62,132,84,99,144,94)(63,121,73,100,133,95)(64,122,74,101,134,96)(65,123,75,102,135,85)(66,124,76,103,136,86)(67,125,77,104,137,87)(68,126,78,105,138,88)(69,127,79,106,139,89)(70,128,80,107,140,90)(71,129,81,108,141,91)(72,130,82,97,142,92), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96)(97,98,99,100,101,102,103,104,105,106,107,108)(109,110,111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130,131,132)(133,134,135,136,137,138,139,140,141,142,143,144)>;

G:=Group( (1,128)(2,129)(3,130)(4,131)(5,132)(6,121)(7,122)(8,123)(9,124)(10,125)(11,126)(12,127)(13,65)(14,66)(15,67)(16,68)(17,69)(18,70)(19,71)(20,72)(21,61)(22,62)(23,63)(24,64)(25,87)(26,88)(27,89)(28,90)(29,91)(30,92)(31,93)(32,94)(33,95)(34,96)(35,85)(36,86)(37,78)(38,79)(39,80)(40,81)(41,82)(42,83)(43,84)(44,73)(45,74)(46,75)(47,76)(48,77)(49,139)(50,140)(51,141)(52,142)(53,143)(54,144)(55,133)(56,134)(57,135)(58,136)(59,137)(60,138)(97,111)(98,112)(99,113)(100,114)(101,115)(102,116)(103,117)(104,118)(105,119)(106,120)(107,109)(108,110), (1,39,109,50,28,18)(2,40,110,51,29,19)(3,41,111,52,30,20)(4,42,112,53,31,21)(5,43,113,54,32,22)(6,44,114,55,33,23)(7,45,115,56,34,24)(8,46,116,57,35,13)(9,47,117,58,36,14)(10,48,118,59,25,15)(11,37,119,60,26,16)(12,38,120,49,27,17)(61,131,83,98,143,93)(62,132,84,99,144,94)(63,121,73,100,133,95)(64,122,74,101,134,96)(65,123,75,102,135,85)(66,124,76,103,136,86)(67,125,77,104,137,87)(68,126,78,105,138,88)(69,127,79,106,139,89)(70,128,80,107,140,90)(71,129,81,108,141,91)(72,130,82,97,142,92), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96)(97,98,99,100,101,102,103,104,105,106,107,108)(109,110,111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130,131,132)(133,134,135,136,137,138,139,140,141,142,143,144) );

G=PermutationGroup([[(1,128),(2,129),(3,130),(4,131),(5,132),(6,121),(7,122),(8,123),(9,124),(10,125),(11,126),(12,127),(13,65),(14,66),(15,67),(16,68),(17,69),(18,70),(19,71),(20,72),(21,61),(22,62),(23,63),(24,64),(25,87),(26,88),(27,89),(28,90),(29,91),(30,92),(31,93),(32,94),(33,95),(34,96),(35,85),(36,86),(37,78),(38,79),(39,80),(40,81),(41,82),(42,83),(43,84),(44,73),(45,74),(46,75),(47,76),(48,77),(49,139),(50,140),(51,141),(52,142),(53,143),(54,144),(55,133),(56,134),(57,135),(58,136),(59,137),(60,138),(97,111),(98,112),(99,113),(100,114),(101,115),(102,116),(103,117),(104,118),(105,119),(106,120),(107,109),(108,110)], [(1,39,109,50,28,18),(2,40,110,51,29,19),(3,41,111,52,30,20),(4,42,112,53,31,21),(5,43,113,54,32,22),(6,44,114,55,33,23),(7,45,115,56,34,24),(8,46,116,57,35,13),(9,47,117,58,36,14),(10,48,118,59,25,15),(11,37,119,60,26,16),(12,38,120,49,27,17),(61,131,83,98,143,93),(62,132,84,99,144,94),(63,121,73,100,133,95),(64,122,74,101,134,96),(65,123,75,102,135,85),(66,124,76,103,136,86),(67,125,77,104,137,87),(68,126,78,105,138,88),(69,127,79,106,139,89),(70,128,80,107,140,90),(71,129,81,108,141,91),(72,130,82,97,142,92)], [(1,2,3,4,5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32,33,34,35,36),(37,38,39,40,41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56,57,58,59,60),(61,62,63,64,65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80,81,82,83,84),(85,86,87,88,89,90,91,92,93,94,95,96),(97,98,99,100,101,102,103,104,105,106,107,108),(109,110,111,112,113,114,115,116,117,118,119,120),(121,122,123,124,125,126,127,128,129,130,131,132),(133,134,135,136,137,138,139,140,141,142,143,144)]])

C2×C6×C12 is a maximal subgroup of   C627C8  C62.15Q8  C6210Q8  C62.247C23  C62.129D4  C6219D4

144 conjugacy classes

 class 1 2A ··· 2G 3A ··· 3H 4A ··· 4H 6A ··· 6BD 12A ··· 12BL order 1 2 ··· 2 3 ··· 3 4 ··· 4 6 ··· 6 12 ··· 12 size 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1

144 irreducible representations

 dim 1 1 1 1 1 1 1 1 type + + + image C1 C2 C2 C3 C4 C6 C6 C12 kernel C2×C6×C12 C6×C12 C2×C62 C22×C12 C62 C2×C12 C22×C6 C2×C6 # reps 1 6 1 8 8 48 8 64

Matrix representation of C2×C6×C12 in GL3(𝔽13) generated by

 12 0 0 0 12 0 0 0 1
,
 12 0 0 0 10 0 0 0 12
,
 8 0 0 0 4 0 0 0 9
G:=sub<GL(3,GF(13))| [12,0,0,0,12,0,0,0,1],[12,0,0,0,10,0,0,0,12],[8,0,0,0,4,0,0,0,9] >;

C2×C6×C12 in GAP, Magma, Sage, TeX

C_2\times C_6\times C_{12}
% in TeX

G:=Group("C2xC6xC12");
// GroupNames label

G:=SmallGroup(144,178);
// by ID

G=gap.SmallGroup(144,178);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-3,-2,432]);
// Polycyclic

G:=Group<a,b,c|a^2=b^6=c^12=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations

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